Math Problem Statement

Find an angle θ with 0° < θ < 360° that has the same sine and cosine values as 10°.

Solution

This question requires you to determine angles θ\theta that share the same sine and cosine values as a given reference angle of 1010^\circ, within the interval 0<θ<3600^\circ < \theta < 360^\circ. Let me break it down for you:

Key Concepts:

  1. Sine Symmetry: The sine of an angle is the same for θ\theta and 180θ180^\circ - \theta. It repeats every 360360^\circ.

    • For sin(10)\sin(10^\circ), the equivalent angle in the interval 0<θ<3600^\circ < \theta < 360^\circ is: 18010=170180^\circ - 10^\circ = 170^\circ
  2. Cosine Symmetry: The cosine of an angle is the same for θ\theta and 360θ360^\circ - \theta. It also repeats every 360360^\circ.

    • For cos(10)\cos(10^\circ), the equivalent angle in the interval 0<θ<3600^\circ < \theta < 360^\circ is: 36010=350360^\circ - 10^\circ = 350^\circ

Final Answers:

  • Sine equivalent: 170170^\circ
  • Cosine equivalent: 350350^\circ

Would you like more details or further assistance? Here are some related questions you could explore:

  1. What are the other symmetrical properties of trigonometric functions on the unit circle?
  2. How do sine and cosine relate to the unit circle visually?
  3. What happens if the angle θ\theta is negative or exceeds 360360^\circ?
  4. How does symmetry in sine and cosine aid in solving equations?
  5. How would these results differ if the angle were given in radians?

Tip: Memorizing key unit circle angles and their symmetries will make these calculations much faster!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Unit Circle
Symmetry of Sine and Cosine

Formulas

sin(θ) = sin(180° - θ)
cos(θ) = cos(360° - θ)

Theorems

Symmetry Properties of Sine and Cosine

Suitable Grade Level

Grades 10-12